In​ March, a family starts saving for a vacation they are planning for the end of August. The family expects the vacation to cost ​$1364. They start with ​$125. At the beginning of each month they plan to deposit 20​% more than the previous month. Will they have enough money for their​ trip? If​ not, how much more do they​ need?

Question

Question content area bottom
Part 1
Select the correct answer below​ and, if​ necessary, fill in the answer box within your choice.
A.
​No, they need ​$
  
enter your response here more. ​(Round up to the nearest​ dollar.)
B.
​Yes, they will have enough money.

Answer 1

To determine whether the family will have enough money for their vacation, let's calculate their total savings for the months of March through August.

1. Initial Savings: $125
2. Expected Cost of Vacation: $1364
3. Monthly Deposits: They will deposit 20% more each month than the previous month.

Let's calculate the deposits for each month:

• Month 1 (March):
  • Deposit = \( x_1 \)
• Month 2 (April):
  • Deposit = \( x_2 = x_1 \times 1.2 \)
• Month 3 (May):
  • Deposit = \( x_3 = x_2 \times 1.2 = x_1 \times 1.2^2 \)
• Month 4 (June):
  • Deposit = \( x_4 = x_3 \times 1.2 = x_1 \times 1.2^3 \)
• Month 5 (July):
  • Deposit = \( x_5 = x_4 \times 1.2 = x_1 \times 1.2^4 \)
• Month 6 (August):
  • Deposit = \( x_6 = x_5 \times 1.2 = x_1 \times 1.2^5 \)

We will also need to recognize that they can choose an initial deposit amount. If we denote the initial deposit in March as \( x_1 \), then their deposits for these six months will be:

• March: \( x_1 \)
• April: \( x_1 \times 1.2 \)
• May: \( x_1 \times 1.2^2 \)
• June: \( x_1 \times 1.2^3 \)
• July: \( x_1 \times 1.2^4 \)
• August: \( x_1 \times 1.2^5 \)

The total amount saved by the end of August becomes:

\[ \text{Total Savings} = 125 + x_1 + x_1(1.2) + x_1(1.2^2) + x_1(1.2^3) + x_1(1.2^4) + x_1(1.2^5) \]

This can be simplified using the formula for the sum of a geometric series, \( S_n = a \frac{(1-r^n)}{(1-r)} \) where \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms.

Here:

• \( a = x_1 \)
• \( r = 1.2 \)
• \( n = 6 \)

The sum of deposits made will then be:

\[ S = x_1 \frac{(1 - 1.2^6)}{(1 - 1.2)} = x_1 \frac{(1 - 2.985984)}{-0.2} \approx x_1 \cdot 9.92992 \]

Thus, adding the initial $125 to the sum of the deposits gives:

\[ \text{Total Savings} = 125 + 9.92992 x_1 \]

To ensure they have enough, this total must be at least $1364:

\[ 125 + 9.92992 x_1 \geq 1364 \]

Isolating \( x_1 \):

\[ 9.92992 x_1 \geq 1364 - 125 \] \[ 9.92992 x_1 \geq 1239 \] \[ x_1 \geq \frac{1239}{9.92992} \approx 124.569 \]

Thus, to know how much money they need for their deposits starting at March, they would need to start depositing at least:

\[ x_1 \approx 125.569 \text{ (rounded up to } 126\text{)} \]

Now, let's check how much they would actually have if they started with the minimum required (126):

After checking with \( x_1 = 126 \):

• March: 126
• April: 151.2
• May: 181.44
• June: 217.728
• July: 261.2736
• August: 313.52832

Total =

\[ 125 + 126 + 151.2 + 181.44 + 217.728 + 261.2736 + 313.52832 \approx 1375.60192 \]

This means they would have:

\( 1375.60192 - 1364 = 11.60192 \)

Thus, they can actually afford their trip with \( x_1 \approx 126 \).

Conclusion

Answer B: Yes, they will have enough money.

Answer 2

wrong

Answer 3

Apologies for the confusion in my previous calculations. Let's go through the problem step by step again to make sure we get the correct outcome.

Given:

• Initial amount = $125
• Vacation cost = $1364
• The monthly deposit increases by 20% starting at \( x_1 \) in March.

Let’s denote the initial deposit (which we will find) as \( x_1 \).

Monthly Deposits:

1. Month 1 (March): Deposit = \( x_1 \)
2. Month 2 (April): Deposit = \( x_1 \times 1.2 \)
3. Month 3 (May): Deposit = \( x_1 \times 1.2^2 \)
4. Month 4 (June): Deposit = \( x_1 \times 1.2^3 \)
5. Month 5 (July): Deposit = \( x_1 \times 1.2^4 \)
6. Month 6 (August): Deposit = \( x_1 \times 1.2^5 \)

Total Savings Calculation:

The total savings by the end of August can be expressed as: \[ \text{Total Savings} = 125 + x_1 + x_1(1.2) + x_1(1.2^2) + x_1(1.2^3) + x_1(1.2^4) + x_1(1.2^5) \]

This can be simplified using the formula for the sum of a geometric series: \[ S_n = a \frac{(1 - r^n)}{(1 - r)} \] where:

• \( a = x_1 \)
• \( r = 1.2 \)
• \( n = 6 \)

Sum of Deposits:

The sum of deposits from March to August is: \[ \text{Deposit Total} = x_1 \frac{(1 - 1.2^6)}{(1 - 1.2)} = x_1 \frac{(1 - 2.985984)}{-0.2} \approx x_1 \cdot \frac{-1.985984}{-0.2} \approx x_1 \cdot 9.92992 \]

Total Savings:

Combining this with the initial savings: \[ \text{Total Savings} = 125 + 9.92992 x_1 \]

Setting up the inequality:

To find out if this is enough for the vacation: \[ 125 + 9.92992 x_1 \geq 1364 \] \[ 9.92992 x_1 \geq 1364 - 125 \] \[ 9.92992 x_1 \geq 1239 \] \[ x_1 \geq \frac{1239}{9.92992} \approx 124.569 \]

Thus, the minimum amount they should start depositing in March must be approximately \( x_1 \geq 125 \) (rounding up as we need to consider whole dollars).

Total Amount if \( x_1 = 125 \):

Let’s calculate the total savings with \( x_1 = 125 \):

• Month 1 (March): \( 125 \)
• Month 2 (April): \( 125 \times 1.2 = 150 \)
• Month 3 (May): \( 150 \times 1.2 = 180 \)
• Month 4 (June): \( 180 \times 1.2 = 216 \)
• Month 5 (July): \( 216 \times 1.2 = 259.2 \)
• Month 6 (August): \( 259.2 \times 1.2 = 311.04 \)

Now summing these:

• Total deposits = \( 125 + 150 + 180 + 216 + 259.2 + 311.04 \approx 1241.24 \)
• Adding the initial savings: \( 125 + 1241.24 = 1366.24 \)

Conclusion:

• If they deposit $125, they will have approximately $1366, which is LESS than $1364.
• The vacation costs $1364.

Therefore, they would not have enough for their trip.

Final Calculation:

• They need approximately: \( 1364 - 1366.24 = 2.24 \)
• Rounding up, they will need at least $3 more.

So the correct response is: A. No, they need $3 more.

Answer 4

round to the nearest dollar

Answer 5

Thank you for your patience! Let's clarify the results with correct calculations and round to the nearest dollar.

Summary of Previous Calculations

• The total cost of the vacation: $1364
• Initial amount saved by the family: $125

Monthly Savings

Let’s denote the initial deposit (in March) as \( x_1 = 125 \).

1. Month 1 (March): Deposit = $125
2. Month 2 (April): Deposit = \( 125 \times 1.2 = 150 \)
3. Month 3 (May): Deposit = \( 150 \times 1.2 = 180 \)
4. Month 4 (June): Deposit = \( 180 \times 1.2 = 216 \)
5. Month 5 (July): Deposit = \( 216 \times 1.2 = 259.2 \)
6. Month 6 (August): Deposit = \( 259.2 \times 1.2 = 311.04 \)

Total Deposits Calculation

Now, summing all the deposits:

• Total deposits \( = 125 + 150 + 180 + 216 + 259.2 + 311.04 \)

Calculating this: \[ = 125 + 150 + 180 + 216 + 259.2 + 311.04 \] \[ = 125 + 150 = 275 \] \[ = 275 + 180 = 455 \] \[ = 455 + 216 = 671 \] \[ = 671 + 259.2 = 930.2 \] \[ = 930.2 + 311.04 = 1241.24 \]

Adding the initial amount: \[ \text{Total Savings} = 125 + 1241.24 = 1366.24 \]

Comparison with Vacation Cost

The total amount they would have is approximately $1366 when rounded down, against the vacation cost of $1364.

Required Amount

• Amount needed for vacation: $1364
• Total they have: $1366

Calculating how much more they need: \[ 1364 - 1366.24 = -2.24 \]

Since they already have enough, and rounding to the nearest dollar:

Final Result

Yes, they will have enough money. They do not need more; in fact, they have approximately $2 left over after the trip expenses.

So, in response to the original question: B. Yes, they will have enough money.